Leading independent schools have published an analysis they claim shows they are outperforming state grammar schools when it comes to adding value in exam results at sixth-form level.
The Headmasters' and Headmistresses' Conference (HMC) looked at statistics for the value added by schools from all sectors between key stages 4 and 5.
But the figures do not take into account "contextual" factors such as pupils' social and ethnic backgrounds, and greater selection in grammars, that could explain the difference.
The top 5 per cent of English schools on the "value added" measure include 16 per cent of HMC's 210 schools but no state grammar schools.
Similarly, the top quarter of English schools on "value added" contained 55 per cent of HMC schools and only 6.7 per cent of the 164 grammar schools in England.
Andrew Grant, HMC chair and head of St Albans School, said: "These figures lay to rest an important myth about HMC schools. They show how much of their pupils' success is down to what the school has added, not how bright the pupils are when they arrive.
"Comparing their performance with selective schools in the maintained sector leads to the conclusion that selection is not the only factor in high performance; it is independence that really makes the difference."
But high value-added scores can be harder to achieve in selective schools because pupils are starting from a higher level and the HMC admits that "many HMC schools are considerably less selective than state grammar schools".
The data used for the analysis is the most comprehensive available from the Government. But it is "value added" at key stage 5 rather than the "contextual value-added" published at key stage 4.
Pupil gender is taken into account. But other variables such as the percentage of pupils eligible for free school meals, which might increase grammar school scores, are not.
Competition between the sectors has increased due to the recession and rising independent school fees.